#Question:
"""Finding the number of poles on the generator if the frequency of the generated voltage is decreased."""
#Variable Declaration:
f=60 #Frequency of ac-generator(in Hertz)
P=6 #Number of poles
#Calculations:
Ns=(120*f)/P
f=20
P=(120*f)/Ns
#Result:
print "The speed of rotation of the generator is %d rpm." %(Ns)
print "If the frequency is reduced to 20Hz,the required number of poles is %d." %(P)
#Question:
"""Finding the distribution factor for a machine."""
from math import radians,degrees,sin
#Variable Declaration:
slots=9 #Number of slots
slot_angle=radians(180/9) #Slot angle(in radians)
#Calculations:
q_a=120.0/degrees(slot_angle)
k_d_a=sin(q_a*(slot_angle/2.0))/(q_a*sin(slot_angle/2.0))
q_b=60.0/degrees(slot_angle)
k_d_b=sin(q_b*(slot_angle/2.0))/(q_b*sin(slot_angle/2.0))
#Result:
print "(a)The distribution factor for a machine for a three phase winding with 120 degrees phase group is %.3f." %(k_d_a)
print "(b)The distribution factor for a machine for a three phase winding with 60 degrees phase group is %.3f." %(k_d_b)
#Question:
"""Finding the speed,the generated emf per phase,and the line emf of a three-phase alternator."""
from math import radians,degrees,sqrt,sin
#Variable Declaration:
phase=3 #Number of phases
f=50 #Frequency rating(in Hertz)
P=20 #Number of poles
slots=180 #Number of slots on the stator
cond_per_slot=8 #Number of conductors per slot
flux=25e-03 #Flux per pole(in Weber)
#Calculations:
Z=slots*cond_per_slot
T=(Z/2)/phase
Ns=(120*f)/P
k_p=1
slots_per_pole=slots/P
slot_angle=radians(180/slots_per_pole)
q=slots_per_pole/phase
k_d=sin(q*(slot_angle/2))/(q*sin(slot_angle/2))
E=4.44*f*flux*T*k_p*k_d
line_emf=sqrt(3.0)*E
#Result:
print "(a)The speed of the alternator is %d rpm." %(round(Ns,0))
print "(b)The rms value of generated EMF per phase is %.2f V." %(E)
print "(c)The line EMF is %.2f V." %(line_emf)
#Question:
"""Finding the voltage regulation for full-load for a three-phase alternator."""
from math import sqrt,cos,acos
#Variable Declaration:
P=600e06 #Power rating of the alternator(in VA)
V_L=22e03 #Rated terminal voltage(in Volts)
sync_imp=0.16 #Synchronous impedance per phase(in Ohms)
res_phase=0.014 #Resistance per phase(in Ohms)
#Calculations:
I_L=P/(sqrt(3)*V_L)
Iph=I_L
V=V_L/sqrt(3)
Vz=Iph*sync_imp
theta=acos(res_phase/sync_imp)
pf=0.8
phi=acos(pf)
alpha=theta-phi
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
vol_reg_a=(E-V)/V
pf=1
phi=acos(pf)
alpha=theta-phi
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
vol_reg_b=(E-V)/V
pf=0.8
phi=acos(pf)
alpha=theta+phi
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
vol_reg_c=(E-V)/V
#Result:
print "(a)For a power factor of 0.8 lagging:"
print "The voltage regulation for full load is %.2f percent." %(vol_reg_a*100)
print "(b)For unity power factor:"
print "The voltage regulation for full load is %.2f percent." %(vol_reg_b*100)
print "(c)For a power factor of 0.8 leading:"
print "The voltage regulation for full load is %.2f percent." %(vol_reg_c*100)
print "\nNote:The voltage regulation for leading power-factor load is negative."
print "It means that on removing the load, the terminal voltage decreases."
#Question:
"""Finding the synchronous reactance per phase and the voltage regulation for a three-phase star-connected alternator."""
from math import acos,sqrt,cos
#Variable Declaration:
V_L=900 #Open-circuit voltage(line to line)(in Volts)
V_L_rated=3.3e03 #Rated voltage of the alternator(in Volts)
I_f=100 #Full-load current(in Amperes)
R=0.9 #Armature Resistance(in Ohm/phase)
#Calculations:
V_oc=V_L/sqrt(3)
I_sc=I_f
sync_imp=V_oc/I_sc
sync_rea=sqrt((sync_imp*sync_imp)-(R*R))
theta=acos(R/sync_imp)
V=V_L_rated/sqrt(3)
Vz=I_f*sync_imp
pf=0.8
phi=acos(pf)
alpha=theta-phi
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
vol_reg_a=(E-V)/V
pf=0.8
phi=acos(pf)
alpha=theta+phi
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
vol_reg_b=(E-V)/V
#Result:
print "The synchronous reactance per phase is %.3f Ohms." %(sync_rea)
print "(a) For a power factor of 0.8 lagging:"
print "The voltage regulation for full load is %.2f percent" %(vol_reg_a*100)
print "(b) For a power factor of 0.8 leading:"
print "The voltage regulation for full load is %.2f percent." %(vol_reg_b*100)
print "\nNote: The voltage regulation for leading power-factor load is negative."
print "It means that on removing the load, the terminal voltage decreases."
#Question:
"""Finding the angle of retard of a synchronous motor."""
from math import sqrt,acos,cos,pi,asin,sin,degrees
from cmath import phase
#Variable Declaration:
Po=9e03 #Power rating of synchronous motor(in Watts)
V_L=400 #Voltage rating of synchronous motor(in Watts)
Zs=0.4+3*1j #Synchronous impedance per phase(in Ohms)
pf=0.8 #Power factor(leading)
effi=0.9 #Efficiency of the motor
#Calculations:
Pin=Po/effi
I_L=Pin/(sqrt(3)*V_L*pf)
I=I_L
phi=acos(pf)
mod_Zs=abs(Zs)
theta=phase(Zs)
V=V_L/sqrt(3)
Er=I*mod_Zs
E=sqrt((V*V)+(Er*Er)+(2*V*Er*cos(pi-(theta+phi))))
E_L=sqrt(3)*E
angle_retard=asin((Er*sin(theta+phi))/E)
#Result:
print "The angle of retard of the rotor is %.2f degrees." %(degrees(angle_retard))
print "The excitation emf E to which the motor has to be excited to give a full-load output at 0.8 leading power factor is %.2f V." %(E_L)
#Question:
"""Finding the line emf generated by the alternator."""
#Variable Declaration:
S=24.0 #Number of slots in the alternator
C=12.0 #Number of conductors per slot
flux=0.1 #Flux per pole(in Weber)
P=4.0 #Number of poles in the alternator
Ns=1500.0 #Synchronous speed of the alternator(in rpm)
#Calculations:
Zph=S*C/3.0
T=Zph/2.0
S_pole=S/P
slot_ang=180.0/S_pole
q=S_pole/3.0
kd=sin((q*radians(slot_ang)/2))/(q*sin(radians(slot_ang/2)))
f=(P*Ns)/120.0
kp=1.0
E=4.44*flux*f*T*kp*kd
E_L=sqrt(3.0)*E
#Result:
print "The line emf generated when the alternator runs at 1500 rpm is %.2f V." %(E_L)
#Question:
"""Finding the net emf induced in the 6 coil in series constituting the alternator winding."""
from math import radians,sin
#Variable Declaration:
q=6.0 #Number of slots per pole per phase
angle=30.0 #Electrical angle between two consecutive slotss(in degrees)
e=10.0 #Emf of each coil(in Volts)
#Calculations:
kd=sin(q*(radians(angle/2.0)))/(q*sin(radians(angle/2.0)))
arith_sum=6*e
Er=kd*arith_sum
#Result:
print "The net emf induced in the six coils in series is %.3f V." %(round(Er,3))
#Question:
"""Finding the number of poles and the current rating of an alternator."""
from math import sqrt,pi
#Variable Declaration:
N=120.0 #Speed of the alternator(in rpm)
f=50.0 #Frequency of the alternator(in Hertz)
VA=100e06 #VA rating of the alternator(in Volt-Ampere)
pf=1.0 #Power factor
V_L=11e03 #Line voltage(in Volts)
effi=0.97 #Efficiency of the alternator
#Calculations:
P=(120.0*f)/N
Po=VA*pf
I_L=VA/(sqrt(3.0)*V_L)
Pin=Po/effi
tor=Pin/(2*pi*N/60.0)
#Result:
print "(a)The number of poles is %d." %(P)
print "(b)The current rating is %.2f A." %(I_L)
print "(c)The input power is %.2f MW." %(round((Pin/1000000),2))
print "(d)The prime-mover torque applied to the genrator shaft is %e Nm." %(tor)
#Question:
"""Finding the percentage regulation for a load."""
from math import atan,radians,acos
#Variable Declaration:
V_L=11e03 #Line voltage of the star-connected alternator(in Volts)
VA=800e03 #Power rating of the alternator(in Volt-Amperes)
Rs=1.5 #Resistance per phase(in Ohms)
Xs=25.0 #Synchronous reactance(in Ohms)
pf=0.8 #Leading power factor
Po=600e03 #Output power(in Watts)
#Calculations:
Vph=V_L/sqrt(3.0)
V=Vph
Iph=Po/(sqrt(3.0)*V_L*pf)
Zs=sqrt((Rs*Rs)+(Xs*Xs))
theta=atan(Xs/Rs)
Vz=Iph*Zs
pf_ang=acos(pf)
alpha=theta+pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
reg=(E-V)/V
#Result:
print "The percentage regulation is %.3f per cent." %(reg*100)
#Question:
"""Finding the full-load voltage regulation of the alternator."""
from math import atan,acos
#Variable Declaration:
V_L=6e03 #Line voltage of the alternator(in Volts)
VA=6000e03 #Power rating of the alternator(in Volt-Amperes)
R=0.2 #Winding resistance per phase(in Ohms)
pf=0.8 #Lagging power factor
V_L_OC=480.0 #Line voltage in open-circuit test(in Volts)
I_f_OC=10.0 #Field current in open-circuit test(in Amperes)
I_L_SC=105.0 #Line current in short-circuit test(in Amperes)
I_f_SC=5.0 #Field current in short-circuit test(in Amperes)
#Calculations:
"""In the short-circuit test,the currents are small compared to the full-load current."""
V=V_L/sqrt(3.0)
I=VA/(3*V)
V_ph_OC=V_L_OC/sqrt(3.0)
"""Since the field current of 5 A gives an armature current of 105 A,a field current of 10 A will give an armature
current of (105*2)=210 A."""
I_ph=I_L_SC*2
Zs=V_ph_OC/I_ph
Xs=sqrt((Zs*Zs)-(R*R))
theta=atan(Xs/R)
Vz=I*Zs
pf_ang=acos(pf)
alpha=theta-pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
reg=(E-V)/V
#Result:
print "The full-load voltage regulation of the alternator at 0.8 lagging power factor is %.2f per cent." %(reg*100)
#Question:
"""Finding the terminal voltage of the generator."""
#Variable Declaration:
f=50.0 #Frequency rating of the generator(in Hertz)
Z_p=96.0 #Number of conductors per phase
flux=0.1 #Flux per pole(in Webers)
Xs=5.0 #Synchronous reactance per phase(in Ohms)
kd=0.96 #Distribution factor for the stator winding
Z_L=10.0 #Load impedance(in Ohms)
#Calculations:
Zs=1j*Xs
kp=1.0
T=Z_p/2.0
E=4.44*f*flux*kp*kd*T
V=E/(1+(Zs/Z_L))
V_L=sqrt(3.0)*abs(V)
#Result:
print "The terminal voltage of the generator is %.2f V." %(V_L)
#Question:
"""Finding the percentage change of voltage."""
from math import atan,radians,acos
#Variable Declaration:
V_L=6.6e03 #Line voltage of the star-connected alternator(in Volts)
VA=1500e03 #Power rating of the alternator(in Volt-Amperes)
Rs=0.5 #Resistance per phase(in Ohms)
Xs=5.0 #Synchronous reactance(in Ohms)
pf=0.8 #Lagging power factor
#Calculations:
Vph=V_L/sqrt(3.0)
V=Vph
Iph=VA/(3.0*Vph)
Zs=sqrt((Rs*Rs)+(Xs*Xs))
theta=atan(Xs/Rs)
Vz=Iph*Zs
pf_ang=acos(pf)
alpha=theta-pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
reg=(E-V)/V
#Result:
print "The percentage change in voltage is %.3f per cent." %(reg*100)
#Question:
"""Finding the terminal voltage and load regulation."""
from math import atan,radians,acos,asin
#Variable Declaration:
V_L=6599.0 #Line voltage of the star-connected alternator(in Volts)
Rs=0.5 #Resistance per phase(in Ohms)
Xs=5.0 #Synchronous reactance(in Ohms)
I=130.0 #Full-load current(in Amperes)
#Calculations:
E=V_L/sqrt(3.0)
Zs=Rs+1j*Xs
theta=atan(Xs/Rs)
Vz=I*abs(Zs)
pf=0.8
pf_ang=acos(pf)
alpha=theta-pf_ang
tor_ang=asin((Vz*sin(alpha))/E)
V_a=(E*cos(tor_ang))-(Vz*cos(alpha))
reg_a=(E-V_a)/V_a
pf=0.6
pf_ang=acos(pf)
alpha=theta+pf_ang
beta=pi-alpha
tor_ang=asin((Vz*sin(beta))/E)
V_b=(E*cos(tor_ang))+(Vz*cos(beta))
reg_b=(E-V_b)/V_b
#Result:
print "(a)The terminal voltage when the power factor is 0.8 lagging is %.2f V." %(V_a)
print " The load regulation is %.2f per cent." %(reg_a*100)
print "(b)The terminal voltage when the power factor is 0.6 leading is %.2f V." %(V_b)
print " The load regulation is %.2f per cent." %(reg_b*100)
#Question:
"""Finding the percentage load regulation."""
from math import acos,atan
#Variable Declration:
VA=1.5e06 #Power rating of the synchronous generator(in Volt-Amperes)
V_L=11e03 #Line voltage(in Volts)
Rs=1.2 #Armature resistance(in Ohms)
Xs=25.0 #Synchronous reactance per phase(in Ohms)
P_L=1.4375e06 #Load to be delivered(in Volt-Amperes)
#Calculations:
V=V_L/sqrt(3.0)
I=P_L/(3*V)
Zs=sqrt((Rs*Rs)+(Xs*Xs))
Vz=I*Zs
pf=0.8
pf_ang=acos(pf)
theta=atan(Xs/Rs)
alpha=theta-pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
reg_a=(E-V)/V
alpha=theta+pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
reg_b=(E-V)/V
#Result:
print "(a)The percentage load regulation for a power factor of 0.8 lagging is %.2f per cent." %(reg_a*100)
print "(b)The percentage load regulation for a power factor of 0.8 leading is %.2f per cent." %(reg_b*100)
#Question:
"""Finding the torque angle of the alternator."""
from math import acos,degrees
#Variable Declaration:
R=0.5 #Effective resistance per phase of the alternator(in Ohms)
V_L=2200.0 #Line voltage of the alternator(in Volts)
I_FL=200.0 #Full-load current(in Amperes)
If=30.0 #Field current(in Amperes)
V_L_oc=1100.0 #Line-to-line voltage on open circuit(in Volts)
pf=0.8 #Lagging power factor
#Calculations:
Zs=V_L_oc/(sqrt(3.0)*I_FL)
theta=acos(R/Zs)
Vz=I_FL*Zs
V=V_L/sqrt(3.0)
pf_ang=acos(pf)
alpha=theta-pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
Po=V*I_FL*pf
tor_ang=theta-acos((Po+(V*V*cos(theta)/Zs))*(Zs/(E*V)))
#Result:
print "The torque angle of the alternator is %.2f degrees." %(degrees(tor_ang))
#Question:
""""Finding the synchronising power of the alternator."""
from math import acos
#Variable Declaration:
P=6.0 #Number of poles
VA=3e06 #Power rating of the alternator(in Volt-Amperes)
Ns=1000.0 #Speed of operation of the alternator(in rpm)
V_L=3.3e03 #Line voltage of the load(in Volts)
pf=0.8 #Lagging power factor
#Calculations:
V=V_L/sqrt(3.0)
I=VA/(sqrt(3.0)*V_L)
IXs=0.25*V
Xs=IXs/I
rotor_dis=radians(1*(P/2.0))
Vz=I*Xs
theta=pi/2
pf_ang=acos(0.8)
alpha=theta-pf_ang
E=sqrt((V*V)+(Vz*Vz)+(2*V*Vz*cos(alpha)))
Psy=(3*E*V*sin(rotor_dis))/(Xs)
tor=(3*Psy)/(2*pi*Ns/60)
#Result:
print "(a)The synchronising power is %.2f kW." %(Psy/1000)
print "(b)The synchronising torque per mechanical degree is %.2f kNm." %(tor/1000)
#Question:
"""Finding the armature current and power factor."""
from math import radians,degrees
from cmath import rect,phase
#Variable Declaration:
V_L=2300.0 #Line voltage of the winding(in Volts)
f=50.0 #Operating frequency of the synchronous motor(in Hertz)
Psh=205.0 #Power delivered by the motor(in Horse Power)
ang=15.0 #Power angle(in degrees)
Xs=11.0 #Synchronous reactance(in Ohms)
effi=0.90 #Efficiency of the motor
#Calculations:
V=V_L/sqrt(3.0)
Psh=Psh*746.0
Pd=Psh/effi
E=(Pd*Xs)/(3.0*V*sin(radians(ang)))
I=(rect(V,0)-rect(E,radians(-ang)))/(1j*Xs)
pf=cos(phase(I))
#Result:
print "(a)The excitation voltage per phase is %.2f V per phase." %(E)
print "(b)The armature current is %.2f A at a phase angle of %.2f degrees." %(abs(I),degrees(phase(I)))
print "(c)The power factor is %.4f leading." %(pf)
#Question:
"""Finding the line current and the power factor."""
#Variable Declaration:
V_L=6600.0 #Line voltage(in Volts)
Xs=20.0 #Synchronous reactance per phase(in Ohms)
Pin=915e03 #Power consumed by the motor(in Watts)
E_L=8942.0 #Induced line emf per phase(in Volts)
#Calculations:
V=V_L/sqrt(3.0)
E=E_L/sqrt(3.0)
I_cos=Pin/(sqrt(3.0)*V_L)
BN=Xs*I_cos
NA=sqrt((E*E)-(BN*BN))
NO=NA-V
Er=sqrt((NO*NO)+(BN*BN))
I_L=Er/Xs
pf=I_cos/I_L
#Result:
print "(a)The line current is %.2f A." %(I_L)
print "(b)The power factor is %.4f leading." %(pf)
#Question:
"""Finding the power factor when the load on the motor increases."""
from math import acos,asin
#Variable Declaration:
V_L=6600.0 #Line voltage(in Volts)
Xs=15.0 #Synchronous reactance per phase(in Ohms)
Pin=500e03 #Power consumed by the motor(in Watts)
pf=0.8 #Leading power factor
Pin_new=800e03 #New Input power(in Watts)
#Calculations:
V=V_L/sqrt(3.0)
Zs=Xs
I_L=Pin/(sqrt(3.0)*V_L*pf)
Er=I_L*Zs
pf_ang=acos(pf)
alpha=(pi/2)-pf_ang
E=sqrt((V*V)+(Er*Er)+(2*V*Er*cos(alpha)))
I1_cos=Pin_new/(sqrt(3.0)*V_L)
sin_tor=(Pin_new*Xs)/(E*V)
AN=V*sin_tor
NB=E-(V*cos(asin(sin_tor)))
Er1=sqrt((AN*AN)+(NB*NB))
I1=Er1/Xs
pf=I1_cos/I1
#Result:
print "The power factor when the load on the motor increases is %.4f leading." %(pf)
#Question:
"""Finding the power factor and kVA rating of the synchronous motor."""
from math import atan,acos
#Variable Declaration:
Si=500e03 #Load supplied by the three-phase system(in Volt-Amperes)
pf_i=0.5 #Lagging power factor of three-phase system
Po=100.0 #Load supplied by the synchronous motor(in Horse-Power)
pf_t=0.9 #Overall lagging power factor
effi=0.87 #Efficiency of the synchronous motor
#Calculations:
Pi=Si*pf_i
Qi=Si*sin(acos(pf_i))
Ps=(Po*746)/effi
Pt=Pi+Ps
St=Pt/pf_t
Qt=St*sin(acos(pf_t))
Qs=Qt-Qi
pf_ang=atan(Qs/Ps)
pf_s=cos(pf_ang)
Ss=sqrt((Ps*Ps)+(Qs*Qs))
#Result:
print "(a)The power faactor of the synchronous motor is %.2f leading." %(pf_s)
print "(b)The kVA rating of the synchronous motor is %.2f kVA." %(Ss/1000.0)